Relevant For...

A sequence is an ordered list of numbers. A series is the sum of the terms of a sequence.

When describing sequences, the following notation is standard:

\[ \{a_n\}_{n=1}^{n=10}, \quad a_n = n^2.\]

The first part of this description, \( \{a_n\}_{n=1}^{n=10} \), could be expanded as a list like this: \( a_1, a_2, a_3, \dots, a_9, a_{10} \). It simply means there are ten terms in the sequence with indices ranging from 1 to 10.

The second part, \( a_n = n^2 \), tells us what each of those terms in the sequence actually is. For example, \( a_4 = 4^2 = 16 \). Thus,

Simple Sequences

A sequence is an ordered set with members called terms. \(_\square\)

Usually, the terms are numbers. A sequence can have infinite terms.

An example of a sequence is \[1,2,3,4,5,6,7,8,\dots.\]

There are different types of sequences. For example, an arithmetic sequence is when the difference between any two consecutive terms in the sequence is the same. So,
\[5, 14, 23, 32, 41,50\]
is an arithmetic sequence with common difference \(9\), first term \(5\), and number of terms \(6.\)

Another type of sequence is a geometric sequence. This is when the ratio of any two consecutive terms in the sequence is the same. For example,
\[2, 6, 18, 54, 162\]
is a geometric sequence with common ratio \(3\), first term \(2\), and number of terms \(5.\)

In a sequence, it is conventional to use the following variables:

\(a\) is the first term in the sequence.

\(n\) is the number of terms in the sequence.

\({ T }_{ n }\) is the \(n^\text{th}\) term in the sequence.

\({ S }_{ n }\) is the sum of the first \(n\) terms of the sequence.

\(d\) is the common difference between any two consecutive terms (arithmetic sequences only).

\(r\) is the common ratio between any two consecutive terms (geometric sequence only).

For example, if a series starts with \(1\) and has a common difference of \(1,\) we have \({ S }_{ n }= \dfrac{n(n + 1)}{2}.\)
Similarly, for the series of squares \(1^2,2^2,3^2,\dots,n^2,\) we have \({ S }_{ n } = \dfrac{n(n + 1)(2n + 1)}{6}.\)
For the series of cubes \(1^3,2^3,3^3,\dots,n^3,\) we have \({ S }_{ n } = \left(\dfrac{n(n+1)}{2}\right)^2.\)

Consider the following sequences:

(1) \(1,3,5,7,9,...\)

(2) the arithmetic sequence \(2,5,8,11,14, ...\)

(3) the geometric sequence \(3,9,27,81, ...\)

In how many of the above sequences have we precisely defined the \(6^\text{th}\) term?